currently:
Wii U
PS4
Xbox One
Ouya
Wii
Xbox 360 (three-traded one but it was broken)
PS3 (two--only one works)
3DS (two)
Vita
PSP (two)
DS (five)
Dreamcast
PS2 (three times--none work, now)
DSi

Previously:
NES
Super NES (mom owns it now)
Gameboy (twice)
Nintendo 64 (twice)
Gameboy Advance
Playstation (twice)
Genesis

Home Consoles -

Playstation, Playstation 2, Playstation 3

Xbox, Xbox 360 (x2), Xbox One

GameCube, Wii U

Handhelds -

Gameboy Color, Gameboy Advanced, 3DS XL

PSP

I still own all of these above. I've been trying to convince my older cousin on giving up his Genesis... which is sadly collecting dust.

2600
nes
snes
genesis
n64
ps1
ps2
GC
ps3
360
ps4

NES
SNES
N64
Gamecube
Wii
Wii U
Sega Master System 2
Mega Drive (aka Genesis in the US)
PS One
PS2
OUYA
Gameboy
Gameboy Color (2)
Gameboy Advance (also 2, one broken)
DS (again, twice)
Game Gear

My dad also had a VC 20 and a TV Game (don't know which one), but he sold them before I could ever play on them.

Oh, and 4 Game & Watch, if these count (don't know their names, 3 of them still work btw)

SNES, Wii (for GC games), Wii U, Xbox One. The list of what I have sold/traded is much longer.

We give away older consoles in my family, so I only have the slightly newer ones, for the most part.

GBA Advance SP (dead though)
Nintendo 3DS
Nintendo 3DS XL
Nintendo 3DS XL (Japanese version)
Wii
Wii U
PS2
PS3
PSP
PS Vita

And my bro has an Xbox One I can borrow without too much trouble (doesn't have enough games for me to want one myself).

In the past though, there was the... (a lot of these systems were shared with other family members though)

Nintendo 64
Gamecube
Gameboy? can't remember
Gameboy Color
Nintendo DS Lite (not shared, had two of them I believe)
Xbox 360

And I now have a high-end gaming-oriented computer, though that's not a console, lol

Wii, X360, 3DS, Wii U and Xbox One. PS4 in the future. (I started off as a Wii Casual at age 11).

Seeing all these posts make me sad, the Xbox One deserves some love :(

Nintendo: SNES, N64, GameCube, Wii, Wii U for home consoles / GB, GBC, GBA, DS, 3DS for handhelds

Sony: PS1, PS2, PS3 for home consoles / PSP, PS Vita for handhelds

Microsoft: Xbox, Xbox 360, Xbox One

Sega: Dreamcast for home consoles / Game Gear for handhelds

Also my PC which I guess doesn't count as a game console, but I do play a lot of games on it :)

Wii U and 3DS this gen/
Wii, DS, and XBOX 360 last gen/
GameCube and GameBoy Advance 2 gens ago
PlayStation 3 gens ago
Genesis 4 gens ago

No doubles.

PS4
PS3
PS2
PS1

PS Vita
PSP

Xbox One
Xbox 360
Xbox

Wii U
Wii
Gamecube
Nintendo 64
SNES
NES

3DS
DS
GBA
Game Boy Colour
Game Boy

Dreamcast
Saturn
Genesis
Master System

Game Gear

Atari 2600/VCS
Intellivision
Coleco Gemini

Home Computers just for kicks
Sinclair 1500
Commodore Vic-20
Commodore 64